課程資訊
 課程名稱 微分幾何一Differential Geometry (Ⅰ) 開課學期 106-1 授課對象 理學院  數學研究所 授課教師 蔡宜洵 課號 MATH7301 課程識別碼 221 U2930 班次 學分 3.0 全/半年 半年 必/選修 選修 上課時間 星期三9(16:30~17:20)星期五3,4(10:20~12:10) 上課地點 天數102天數102 備註 總人數上限：80人 Ceiba 課程網頁 http://ceiba.ntu.edu.tw/1061MATH7301_ 課程簡介影片 核心能力關聯 核心能力與課程規劃關聯圖 課程大綱 為確保您我的權利,請尊重智慧財產權及不得非法影印 課程概述 This class will go through the following topics in a year. 1. Motivations for manifolds; concept of manifolds; vector fields, tangent vectors, 1-forms 2. Pull-back; push-forward; submanifolds 3. Whitney Embedding theorem; immersions, submersions 4. Tangent bundles; tensors; antisymmetric tensors 5. Flow; one parameter subgroup 6. Brackets, Lie derivative, computation via coordinates 8-10. Differential forms, exterior derivative, tensor operations, contractions; Interior product, Cartan’s formula, classical vector analysis, Stokes theorem for differential forms 11-12. Frobenius theorem, Poincare Lemma, De Rham cohomology, differential form version of Frobenius, applications to PDE and geometry 13. Riemannian manifolds, Euclidean case, covariant derivative 14. Riemannian metric, Riemann-Christoffel symbols, parallelism, geodesic 15. Exponential mapping, normal coordinates 16. Convex neighborhoods; two proofs 17. Riemannian curvature, sectional curvature, Ricci curvature 18. Bianchi identity, hypersurface 19. 1st fundamental forms, 2nd fundamental forms determine hypersurfaces up to isometry 20. Variation formula, Gauss lemma 21. Hopf-Rinow theorem 22-24. Jacobi fields, 2nd variation, Jacobi equation, conjugate points, minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs 25-27. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius 28-29. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields 30. Cartan-Ambrose-Hicks Theorem 31. Miscellaneous a) flows and transformations b) Killing vector fields c) volume element and divergence d) Ricci curvature and volume growth e) 2nd Bianchi identity applied to Einstein manifolds f) Cut locus, injectivity radius, Klingenberg’s lemma References for 1st semester: 1. Do Carmo: Riemannian geometry (together with his book on “curves and surfaces”) 2. Gallot-Hulin-Lafontaines: Riemannian geometry 3. Helgason: Differential geometry, Lie groups and symmetric spaces 4.Hicks: Notes on Differential geometry 5. Cheeger-Ebin: Comparison theorems in Riemannian geometry 課程目標 待補 課程要求 待補 預期每週課後學習時數 Office Hours 參考書目 待補 指定閱讀 待補 評量方式(僅供參考)
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